# Solving Vectors & Surfing Problem at Waikiki

• Eugen
In summary: But the water is not moving in the x-y plane. Only the waves and the surfer are moving in the x-y plane. Waves are a deflection in the z direction.
Eugen
Hello,

Here's the problem:

Suppose we orient the x-axis of a two-dimensional coordinate system along the beach at Waikiki. Waves approaching the beach have a velocity relative to thTe shore given by v = (1.3 m/s)y. Surfers move more rapidly than the waves, but at an angle to the beach. The angle is chosen so that the surfers approach the shore with the same speed as the waves. (a) If a surfer has a speed of 7.2 m/s relative to the water, what is her direction of motion relative to the positive x axis? (b) What is the surfer’s velocity relative to the wave? (c) If the surfer’s speed is increased, will the angle in part (a) increase or decrease? Explain.

The problem says that "the surfers approach the shore with the same speed as the waves". I take this to mean "the surfers velocity relative to the shore is the same as the waves velocity relative to the shore: 1.4 m/s". I think that's impossible because the surfers velocity relative to the shore is the waves velocity relative to the shore (1.4 m/s) + the surfers velocity relative to the water (7.2 m/s). Not matter what the angle of the surfers motion is, their speed relative to the shore will be much larger than 1.4 m/s.
Am I missing something?

Eugen said:
Hello,

Here's the problem:

Suppose we orient the x-axis of a two-dimensional coordinate system along the beach at Waikiki. Waves approaching the beach have a velocity relative to thTe shore given by v = (1.3 m/s)y. Surfers move more rapidly than the waves, but at an angle to the beach. The angle is chosen so that the surfers approach the shore with the same speed as the waves. (a) If a surfer has a speed of 7.2 m/s relative to the water, what is her direction of motion relative to the positive x axis? (b) What is the surfer’s velocity relative to the wave? (c) If the surfer’s speed is increased, will the angle in part (a) increase or decrease? Explain.

The problem says that "the surfers approach the shore with the same speed as the waves". I take this to mean "the surfers velocity relative to the shore is the same as the waves velocity relative to the shore: 1.4 m/s". I think that's impossible because the surfers velocity relative to the shore is the waves velocity relative to the shore (1.4 m/s) + the surfers velocity relative to the water (7.2 m/s). Not matter what the angle of the surfers motion is, their speed relative to the shore will be much larger than 1.4 m/s.
Am I missing something?

Welcome to the PF.

(In the future, please fill out the Homework Help Template you are provided when posting a schoolwork question. It makes it easier to answer your questions)

The surfer is moving toward the shore at the same speed as the wave (in the y direction). The surfer is moving in the x direction parallel to the shore with a different speed, and the vector sum of the two is the 7.2m/s with respect to a fixed spot (attached to the Earth).

BTW, in your problem statement the y-speed was stated as 1.3m/s, and then it looks like you changed it to 1.4m/s?

Thanks for answering. :) It's 1.3 m/s all the time...

berkeman said:
The surfer is moving toward the shore at the same speed as the wave (in the y direction). The surfer is moving in the x direction parallel to the shore with a different speed, and the vector sum of the two is the 7.2m/s with respect to a fixed spot (attached to the Earth).

This seems clear enough, but the problem says: "If a surfer has a speed of 7.2 m/s relative to the water,...". So, the 7.2 m/s speed is relative to the water, not to the shore. If the surfer moves parallel to the beach with 7.2 m/s (relative to the water) he can't move with 1.3 m/s relative to the beach. This is what I don't understand.

Eugen said:
Thanks for answering. :) It's 1.3 m/s all the time...
This seems clear enough, but the problem says: "If a surfer has a speed of 7.2 m/s relative to the water,...". So, the 7.2 m/s speed is relative to the water, not to the shore. If the surfer moves parallel to the beach with 7.2 m/s (relative to the water) he can't move with 1.3 m/s relative to the beach. This is what I don't understand.

But the water is not moving in the x-y plane. Only the waves and the surfer are moving in the x-y plane. Waves are a deflection in the z direction.

If there is a beach ball floating on the water (no wind) as the surfer goes by, does the beach ball move toward the shore? (Well, there is a small oscillation in the y-direction, but overall it does not move).

If the water is not moving, then the surfer's speed relative to the water should be the same as his speed relative to the shore (vector sum of the waves speed and surfer's horizontal speed). Or that is not so?
Then how can it be that " the surfers approach the shore with the same speed as the waves."

Eugen said:
If the water is not moving, then the surfer's speed relative to the water should be the same as his speed relative to the shore (vector sum of the waves speed and surfer's horizontal speed). Or that is not so?

Yes, his V(x,y) is the same with respect to the still water and the shore.

Eugen said:
Then how can it be that " the surfers approach the shore with the same speed as the waves."

His V(y) is the same as the waves' V(y) while he is riding the wave into shore.

His motion V(x,y) has two components. One component is 1.3m/s in the y direction, and his x component combines vectorially to make his overall velocity vector V(x,y) = 7.2m/s with respect to the still water and still Earth. Can you write the vector equation that results in the final 7.2m/s?

Yes, that equation is Vs,sh = Vw,sh + Vs,w where w = waves, sh = shore, s = surfer.
Vs,sh = 7.2 m/s and Vw,sh = 1.3 m/s.
But I still don't understand how can the surfer approach the shore with the same speed as the waves. The surfer's speed relative to the shore is 7.2 m/s. The waves speed relative to the shore is 1.3 m/s.

Last edited:
Eugen said:
Yes, that equation is Vs,sh = Vw,sh + Vs,w where w = waves, sh = shore, s = surfer.
Vs,sh = 7.2 m/s and Vw,sh = 1.3 m/s.
But I still don't understand how can the surfer approach the shore with the same speed as the waves. The surfer's speed relative to the shore is 7.2 m/s. The waves speed relative to the shore is 1.3 m/s.

Just sketch the vectors and add them.

berkeman said:
Just sketch the vectors and add them.
I can't, because I don't know how to find out the x and y components of Vs,w and Vs,sh.

Eugen said:
I can't, because I don't know how to find out the x and y components of Vs,w and Vs,sh.

Draw an x-y coordinate system, with the origin at the shore (sand is positive y and water is negative y). Positive x is to the right and positive y is up.

Then draw the wave at y = -30m (or whatever). That's a horizontal line going from left to right. Draw a dot at x = 0 on that wave line, and that's your surfer.

Draw a vertical velocity vector in the y direction of 1.3m/s. That's the velocity of the wave and surfer in the y direction (towards the beach). Then draw a longer velocity vector in the +x direction along the wave line -- that is the component of the surfer's velocity parallel to the wave. Then draw a resultant vector from adding the x and y components of his velocity -- it will point up and to the right, and will have a magnitude of 7.2m/s. You should be able to use the Pythagorean theorem to figure out the x component of the surfer's velocity from this sketch...

I didn't realize that the surfer's movement relative to waves is perpendicular on the waves movement. Is this related to the surfer approaching the shore with the same speed as the waves? If the surfer moved at some angle with respect to the waves, wouldn't he approach the shore at the same speed as the waves?
Put this way, the problem is very simple. The answer at point a) is 10 °.

berkeman

## 1. What are vectors and how are they used in surfing at Waikiki?

Vectors are quantities that have both magnitude and direction, and they are used in surfing at Waikiki to describe the movement of waves and the motion of surfers. By understanding vectors, surfers can predict the direction and speed of incoming waves and adjust their movements accordingly.

## 2. How do you solve for vectors in surfing at Waikiki?

To solve for vectors in surfing at Waikiki, surfers use mathematical equations and diagrams to break down the movement of waves and surfers into its individual components, such as magnitude and direction. This allows them to accurately predict the behavior of the waves and make strategic surfing decisions.

## 3. What are some common problems that can arise when solving vectors in surfing at Waikiki?

Some common problems that can arise when solving vectors in surfing at Waikiki include accounting for variables such as currents, wind, and other surfers, as well as the constantly changing conditions of the ocean. It is important for surfers to regularly reassess and adjust their calculations as they ride the waves.

## 4. What techniques can be used to improve vector calculations in surfing at Waikiki?

Some techniques that can be used to improve vector calculations in surfing at Waikiki include practicing regularly to improve one's understanding of the ocean and its movements, using technology such as GPS trackers and wave sensors, and seeking guidance from experienced surfers and coaches.

## 5. How can understanding vectors enhance the surfing experience at Waikiki?

Understanding vectors can enhance the surfing experience at Waikiki by allowing surfers to make more informed and strategic decisions, leading to a more enjoyable and successful surfing session. It can also improve safety by helping surfers anticipate and avoid potential hazards in the water.

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